In this section, we make a short overview of the Galois theory of linear difference equations.
For more details on what follows, we refer to [1]}.
A difference ring is a commutative ring \(\mathbf {R}\)  equipped with an automorphism \(\phi : \mathbf {R}\rightarrow \mathbf {R}\) . We denote by \(C_{\mathbf {R}}:= \lbrace c\in \mathbf {R}| \phi (c)=c\rbrace \) , the constants of the difference ring \((\mathbf {R},\phi )\) . A difference ideal \(I\)  of a difference ring \((\mathbf {R},\phi )\)  is an ideal such that \(\phi (I)\subset I\) . If \(\mathbf {R}\)  is a field, we call \((\mathbf {R},\phi )\)  a difference field. In this case, \(C_{\mathbf {R}}\)  is a field, the field of constants.
With an abuse of notations, we will often denote by \(\mathbf {R}\)  the difference ring \((\mathbf {R},\phi )\) .
